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Abstract

Absolute flatness of three silicon plane mirrors have been measured by a three-intersection
method based on the three-flat method using a near-infrared interferometer. The interferometer
was constructed using a near-infrared laser diode with a 1,310-nm wavelength light
where the silicon plane mirror is transparent. The height differences at the coordinate
values between the absolute line profiles by the three-intersection method have been
evaluated. The height differences of the three flats were 4.5 nm or less. The three-intersection
method using the near-infrared interferometer was useful for measuring the absolute
flatness of the silicon plane mirrors.

Keywords:

Background

High-precision measurements of surface flatnesses are important in the development
of optical devices. In flatness testing, interferometry with a standard flat is used
for high-precision measurements. In a measurement with a standard flat, the measurement
accuracy is mainly determined using the figure of the standard flat. The three-flat
method by interferometry is commonly used to measure the flatness of standard flat
surfaces for high-precision interferometers. This method allows others to measure
the absolute line profile, and its importance is widely accepted [1-4]. The absolute testing of optical flats has been discussed by a rotation-shift method
[5].

High-grade flats are required for interferometry with a standard flat because the
accuracy is critically dependent on the figure. Recently, flattened silicon surfaces
on the nanometer scale have been prepared [6-8]. A silicon flat is expected to be one of the standard flats. The absolute line profile
of the silicon mirror cannot be measured by the three-flat method when a visible light
is used. To measure the absolute line profile of the silicon mirror by the three-flat
method, an interferometer with a light source where the silicon mirror is transparent
must be constructed, and only three silicon mirrors are used to measure the absolute
line profiles. However, the absolute line profile measurement of the silicon mirror
with a near-infrared light has not been carried out using only silicon mirrors. A
near-infrared Fizeau interferometer with a 1.55-μm wavelength laser diode has been
developed to improve the fringe contrast for large surface roughness. However, a near-infrared
interferometer using a shorter wavelength has not been tested [9].

The authors constructed an interferometer using a near-infrared laser diode with a
1,310-nm wavelength light where the silicon plane mirror is transparent. They also
measured the absolute line profiles of three silicon plane mirrors for standard flats
through the use of the three-flat method by near-infrared interferometry [10]. In this paper, the authors describe a method for measuring the absolute flatness
of silicon mirrors by a three-intersection method and determine the absolute shapes
using a near-infrared interferometer. To evaluate the precision of the absolute flatness
measurements, the authors examine the height differences in the absolute shapes.

Methods

Figure 1 shows a schematic diagram of the near-infrared interferometer. The near-infrared
interferometer was built based on the Fizeau interferometer. Figure 2 shows a photograph of the near-infrared interferometer. The near-infrared laser diode
(FOL13DDRC-A31, Furukawa Electric Co., Ltd., Chiyoda-ku, Tokyo, Japan) with a 1,310-nm
peak wavelength light where the silicon plane mirror is transparent, was used as a
light source. The typical peak wavelength of the laser light was 1,310 nm. The temperature
dependence of the peak emission wavelength was 0.09 nm/°C. The ambient temperature
fluctuation during the measurements by the three-flat method was within 0.1°C. The
temperature of the laser diode was within 0.1°C. The wavelength fluctuation was estimated
to be 0.009 nm from the temperature dependence and fluctuation. The output light from
the near-infrared light source was expanded to the necessary size. A parallel light
was provided using the collimator and perpendicularly incident on the reference and
detected surfaces. The reference and detected surfaces were placed almost parallel,
and the distance between them was approximately 24 mm. The light was divided into
two waves on the reference surface. One of the waves was reflected on the surface
and the other passed through it. The wave passing through the reference surface was
reflected on the detected surface. The two reflected waves passing through the imaging
lens interfered and formed interferograms. The image of the interferogram was put
into a personal computer with a near-infrared charge-coupled device (CCD) camera (C5840,
Hamamatsu Photonics K. K., Hamamatsu, Shizuoka, Japan). The CCD camera had a high
sensitivity to wavelengths from 400 to 1,650 nm. The signal of the CCD camera output
was converted to a 10-bit digital signal using a video analog-to-digital converter.
The 32 digital signals were accumulated on a computer with a software (LabVIEW, National
Instruments Corporation, Austin, TX, USA) designed to obtain the average. The first
10 digits of the average signal were chosen as the measured value of the interferogram
intensity.

Figure 3 shows a typical intensity map of an interferogram. The distance between the reference
and detected surfaces varied by an interval of λ/12 to λ/2 with a phase shift stage, and interferograms were recorded at equal intervals of
the shifted distance using the CCD camera. The phase shift stage which was composed
of elastic hinges and a piezoelectric actuator traveled in a straight line. In order
to reduce the effects of environmental vibration or temperature drift on the measurement
precision, the reference surface was moved stepwise for the phase shift, and the intensities
of the interferograms were averaged on a platform at each step. The reference surface
was moved by accelerating or decelerating the drive with the phase shift stage under
low acceleration just after starting or before stopping to avoid the drift of the
reference surface caused by vibration. Environmental vibration was attenuated using
an active vibration-isolated table (AVI-350M, Herz Co., Ltd., Yokohama, Kanagawa,
Japan). The acceleration of the environmental vibration was approximately 2 mgal.
Both the reference and detected surfaces were silicon plane mirror surfaces. The silicon
plane mirror was a square plate with polished surfaces on both sides. To prevent interference
by the reflected light from the back surface of the reference or the detected surface,
a wedge was formed on the back surface of the silicon plane mirrors. The designed
width, thickness, wedge angle, and azimuth angle of the wedge were 50.0 mm, 10.0 mm,
0.28°, and 22.5°, respectively. The silicon plane mirrors were polished with a magnetorheological
finishing (MRF) [11], and the flatnesses were 30 nm or less. The silicon plane mirror was supported at
six points on the sample holder which was fixed on the phase shift stage, and the
mirror was supported at three points on the back surface, two points on the undersurface,
and one point on the side surface.

From the interferogram intensities at each pixel site of the CCD camera, the initial
phase of each pixel site was calculated by 6 + 1-sample algorithm [12]. Figure 4 shows the sampling for the 6 + 1-sample algorithm by the following equation:

The relative heights of the reference and detected surface were calculated from the
initial phases and the wavelength. Three silicon plane mirrors (A, B, and C flats)
were combined in pairs with different positional combinations (transmission reference
A and detected B, A and C, and B and C) in the interferometer and used for calculation
of the absolute line profile of each silicon plane mirror by the three-flat method
[2]. The absolute line profile could be measured only along a vertical center line on
the reference and detected flats. The B flat in the combination B and C was rotated
around the vertical center line compared to the B flat in the combination A and B.
The position of the center and the direction of the center line on the detected flat
were adjusted to be the same as those on the reference flat within 1 pixel of the
CCD camera (which has 640 × 480 pixels). One pixel corresponds to 107 μm on the flat.

Figure 5 shows the arrangement of the reference and detected flats in absolute flatness measurements
by the three-intersection method. Both rotating and shifting were used to eliminate
an indeterminate term that equated to a twisted surface [13]. In the three-intersection method, the reference flat (the B flat) was rotated around
the z-axis, or the detected flat (the C flat) was shifted from the combination B and C
flats in the three-flat method. By rotating the reference flat 90 or -90° clockwise
around the z-axis, as shown in Figure 5a,b, the absolute line profile could be measured only along a diagonal or another
diagonal line on the reference and detected flats. By shifting the detected flat to
y = -20.00 mm or x = -20.00 mm using the XZ stage (FS-1100PXZ, SIGMA TECH. CO., LTD., Hanno, Saitama,
Japan), as shown in Figure 5c,d, the absolute line profile could be measured only along a line at y = 10.0 mm or x = 10.0 mm. Figure 5 shows the test configurations in absolute flatness measurements by the three-intersection
method. An absolute line profile could be measured only along a rotation axis on the
reference or the detected flat by the three-flat method. Figure 6a,b,c shows the configuration of the rotation axis on a diagonal, another diagonal
and a line at y = 10.0 mm, respectively. Heights of the three absolute profiles along the three axes
were adjusted to be zero at three intersections indicated by solid circles in Figure 6c. Five absolute line profiles along the rotation axes parallel to the y-axis were measured at x = -10.0, -5.0, 0.0, 5.0, and 10.0 mm in Figure 6d. The height of each profile was adjusted to be the same as that of the profiles
at the two intersections indicated by solid circles for y = 10.0 mm, one diagonal or for y = 10.0 mm, and another diagonal. Thus, an absolute flatness could be measured by
the three-intersection method.

Figure 5.Arrangement of the reference (lower left) and detected (upper right) flats in the
three-intersection method. For (a) rotation axis on diagonal, (b) another diagonal, (c) line at y = 10.0 mm, and (d) line at x = 10.0 mm.

Results and discussion

Figure 7 shows the relative line profiles of the reference and detected surfaces along the
vertical center line. The relative line profiles were calculated from a set of interferograms
by the 6 + 1-sample algorithm for the one phase-shifting interval of λ/6. The y-axis is the dimension of the measured length. The length was 30.0 mm. The relative
line profiles were calculated for eight measurements. The inclination of the reference
and detected surfaces was removed by applying the least-squares method. The peak-to-valley
(PV) values of the relative line profiles in Figure 7a,b,c are approximately 22, 18, and 24 nm, respectively.

Figure 7.Relative line profiles along a vertical center line. For (a) A and B, (b) A and C, and (c) C and B flats.

Figure 8 shows the height difference between the relative line profiles and the mean value.
The height difference was calculated by subtracting one measured profile from the
mean value. The PV values of eight height differences were 1.5 nm at minimum and 4.7
nm at maximum. When there was a peak in relative line profiles for one combination
of two flats, the corresponding peak could appear in all absolute line profiles of
A, B, and C flats in the three-flat method. The root-mean-square (RMS) values of the
relative line profiles in Figure 8a,b,c were 0.41, 0.48, and 0.35 nm, respectively. The repeatability of the PV measurements
of a commercial interferometer using a visible light of 632.8-nm wavelength was better
than λ/300. It is necessary to compare the same specification between the near-infrared
and visible light interferometers.

Figure 8.Height differences between relative line profiles and mean values. For (a) A and B, (b) A and C, and (c) C and B flats.

Figure 9 shows the absolute line profiles of each silicon plane mirror along the vertical
center line at x = 0.0 mm. The relative line profiles were calculated for eight measurements, and
the absolute line profiles were calculated for each measurement. The PV values of
the absolute line profiles in Figure 9a,b,c were 15.7, 18.4, and 20.6 nm, respectively. The RMS values of the absolute line
profiles in Figure 9a,b,c were 3.0, 3.4, and 3.1 nm, respectively. In Figure 9a,b,c, surface waves are observed. The pitch of the surface waves were approximately
0.75 mm. This suggests that the pitch reflects the feed of the MRF polishing.

Figure 10 shows the absolute shapes of flats by the three-intersection method. Height differences
at the x-y coordinate values (-5, 5) and (5, 5) indicated by open circles in Figure 6d are shown in Table 1. The height differences of the three flats are 4.5 nm or less. The height difference
was due to height differences between the relative line profiles and the mean value.
This result suggests that the absolute flatness of surfaces can be measured by the
three-intersection method by near-infrared interferometry.

Conclusions

The authors measured the absolute flatness of three silicon plane mirrors with the
three-intersection method using the near-infrared interferometer. The height differences
at the x-y coordinate values have been examined to evaluate the precision of the absolute flatness
measurement. The height differences of the three flats were 4.5 nm or less. The absolute
flatness of the surfaces may be measured through the use of the three-intersection
method by near-infrared interferometry. This study represents an initial step toward
the measurement of flattened silicon surfaces using a near-infrared interferometer.

Abbreviations

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

JU proposed a three-intersection method and analyzed the data. YH carried out the
experiments of the three-intersection method using a near-infrared interferometer.
NA fabricated the near-infrared interferometer. KK participated in the sample preparations.
KA investigated the measurement accuracy. MM gave the final approval of the version
to be published. All authors read and approved the final manuscript.

Acknowledgments

The work was supported in part by a Grant-in-Aid for Scientific Research (C) 23560123
from Japan Society for the Promotion of Science.